Grassmannian quantum cohomology in the infinite limit and total positivity

Abstract: The theory of total positivity was shown by Lusztig to be intrinsically linked to the canonical basis with its positivity properties. When we restrict ourselves to studying total positivity just for the set of lower-triangular unipotent Toeplitz matrices, say in type $A$, then there is a similar link with the quantum cohomology rings of flag varieties and the Schubert bases and their positivity properties. Namely, this builds on a theory of Dale Peterson that gives a uniform Lie-theoretic description of all of the quantum cohomology rings $qH<sup>*(G/P)$. In a precursor to this paper, the Schubert basis and quantum parameters in $qH<sup>*(SL_n/B)$, which restrict to positive-valued functions on totally positive Toeplitz matrices, were analysed with respect to their limiting behaviour as $n\to\infty$, uncovering a novel connection with the classical Edrei theorem on parametrising the infinite totally positive Toeplitz matrices. In this paper we study the Grassmannian case, using the conventions from the $SL_{n}/B$ setting as a guide, and we determine the quantum parameter and Schubert class asymptotics in different scenarios. Along the way, we obtain a new interpretation of the strange duality involution on the localised quantum cohomolgy ring of the Grassmannian. Finally, we prove an asymptotic formula for quantum parameters in a partial flag setting, and we furthermore formulate some conjectures concerning partial flag varieties and related quantum cohomology asymptotics.

• The paper classifies limits of totally positive Peterson Toeplitz varieties as n approaches infinity, yielding polynomial, reciprocal, or exponential generating functions for fixed codimension, fixed rank, and balanced Grassmannians.
• It proves that limiting Schubert classes are Schur-polynomial evaluations, specifically representation-theoretic dimensions after specialization, and links quantum parameters to Schoenberg parameters through nth-root asymptotics.
• It identifies Postnikov–Hengelbrock strange duality with a transposition-based involution on the Peterson variety and proposes partial-flag conjectures extending the Grassmannian and Edrei–Schoenberg framework.

Context and motivation

This paper, by Inès Chung-Halpern and Konstanze Rietsch, studies the asymptotic behaviour, as $n \to \infty$, of the totally positive parts of the Peterson Toeplitz varieties associated to Grassmannians $Gr(k,n)$, and relates these limits to the classical Edrei–Schoenberg parametrisation of infinite totally nonnegative Toeplitz matrices. The work builds on a precursor paper by Rietsch (Rietsch, 29 Sep 2025), which established analogous results for full flag varieties $SL_n/B_n$: there, Schubert classes and $n$-th roots of quantum parameters were shown to converge to evaluations of Schubert polynomials at reciprocals of Schoenberg parameters, and to ratios of consecutive Schoenberg parameters, respectively.

The underlying bridge between quantum cohomology and total positivity is Dale Peterson's theory, which identifies the localised quantum cohomology ring $qH^*(SL_n/P)[q_{n_1}^{-1},\dots,q_{n_k}^{-1}]$ with the coordinate ring $\mathbb{C}[X_P]$ of a "Peterson Toeplitz variety" $X_P$ — an intersection of the stabiliser subgroup $X = U_-^F$ of the standard principal nilpotent with a Schubert cell. Under this isomorphism, Schubert classes correspond to quotients of minors $\mathfrak{S}_P^w = \Delta_k^w/\Delta_k$, and Rietsch's earlier work established that the locus where all Schubert classes take positive values coincides exactly with the totally positive part $X_P(\mathbb{R}_{>0})$. The paper's central question is: which infinite totally nonnegative Toeplitz matrices arise as limits of elements of $Gr(k,n)$0 for the various Grassmannian parabolics, and what are the limits of the corresponding Schubert class functions?

Conventions and comparison of Peterson presentations

A substantial preliminary contribution is a careful reconciliation of two versions of Peterson's theorem. The first (Theorem 1 in the paper) identifies $Gr(k,n)$1 with $Gr(k,n)$2, sending $Gr(k,n)$3 to the minor quotient $Gr(k,n)$4; the second identifies it with the coordinate ring of the open stratum $Gr(k,n)$5 of the Peterson variety $Gr(k,n)$6 inside the flag variety, sending $Gr(k,n)$7 to a function $Gr(k,n)$8. The authors construct an explicit isomorphism $Gr(k,n)$9, $SL_n/B_n$0, where $SL_n/B_n$1 is Berenstein–Zelevinsky's positivity-preserving involutive anti-automorphism, and prove via a weight-space argument on Lusztig tori that $SL_n/B_n$2 for all Grassmannian permutations $SL_n/B_n$3. This comparison result underpins everything that follows, including the new interpretation of strange duality discussed below.

Asymptotics for fixed codimension: $SL_n/B_n$4

Fixing $SL_n/B_n$5, the totally positive part $SL_n/B_n$6 is one-dimensional, parametrised by $SL_n/B_n$7 via the map $SL_n/B_n$8, where $SL_n/B_n$9 is the set of "right-most" $n$0-th roots of $n$1 and $n$2 is the Toeplitz matrix whose generating polynomial factors as $n$3. The first main result shows that any convergent sequence $n$4 with $n$5 has a limit whose generating function is necessarily $n$6 for some $n$7, and conversely every such limit arises. The parameter $n$8 is recovered from the quantum parameter by

$n$9

using the identity $qH^*(SL_n/P)[q_{n_1}^{-1},\dots,q_{n_k}^{-1}]$0 proved via the symmetric presentation $qH^*(SL_n/P)[q_{n_1}^{-1},\dots,q_{n_k}^{-1}]$1 of $qH^*(SL_n/P)[q_{n_1}^{-1},\dots,q_{n_k}^{-1}]$2 and the curve $qH^*(SL_n/P)[q_{n_1}^{-1},\dots,q_{n_k}^{-1}]$3 of $qH^*(SL_n/P)[q_{n_1}^{-1},\dots,q_{n_k}^{-1}]$4-th roots configurations. Note the sharp contrast with the full flag case: here the limiting generating function is always a polynomial with all roots equal, because the individual entries $qH^*(SL_n/P)[q_{n_1}^{-1},\dots,q_{n_k}^{-1}]$5 can only depend on finitely many parameters — a structural constraint absent when all $qH^*(SL_n/P)[q_{n_1}^{-1},\dots,q_{n_k}^{-1}]$6 are nonzero.

For Schubert classes, the key observation is a stability lemma: for fixed $qH^*(SL_n/P)[q_{n_1}^{-1},\dots,q_{n_k}^{-1}]$7, the function $qH^*(SL_n/P)[q_{n_1}^{-1},\dots,q_{n_k}^{-1}]$8 is independent of $qH^*(SL_n/P)[q_{n_1}^{-1},\dots,q_{n_k}^{-1}]$9 as a polynomial in the entries $\mathbb{C}[X_P]$0. Combining this with the Jacobi–Trudi identity and a combinatorial bijection between semistandard Young tableaux, the authors prove:

$\mathbb{C}[X_P]$1

where $\mathbb{C}[X_P]$2 is the conjugate partition and $\mathbb{C}[X_P]$3 is a Schur polynomial in $\mathbb{C}[X_P]$4 variables. Since $\mathbb{C}[X_P]$5 equals the dimension of the irreducible $\mathbb{C}[X_P]$6-representation $\mathbb{C}[X_P]$7, the limit carries direct representation-theoretic content. This is the precise Grassmannian analogue of the full flag formulas involving $\mathbb{C}[X_P]$8.

Asymptotics for fixed rank: $\mathbb{C}[X_P]$9

When instead $X_P$0 is fixed, the roles of zeros and poles interchange. The authors first establish a symmetry map $X_P$1, given concretely by $X_P$2 or equivalently $X_P$3, which realises the duality $X_P$4 swapping $X_P$5 and $X_P$6 at the level of Toeplitz varieties, and preserves total positivity with identical parameter $X_P$7. Applying this symmetry yields:

$X_P$8

and the Schubert class limits become $X_P$9 in $X = U_-^F$0 variables — now indexed by $X = U_-^F$1 itself rather than its conjugate. These results are directly analogous to the full flag asymptotics $X = U_-^F$2 for the torus parameters, confirming that the Grassmannian formulas are specialisations of a uniform pattern governed by the Weyl-parabolic structure.

The middle-dimensional case $X = U_-^F$3 and exponential generating functions

The most striking result concerns $X = U_-^F$4, where both $X = U_-^F$5 and $X = U_-^F$6 grow. Using a Gaussian $X = U_-^F$7-binomial expansion with $X = U_-^F$8 and the asymptotics of $X = U_-^F$9, the authors show that sequences $\mathfrak{S}_P^w = \Delta_k^w/\Delta_k$0 with $\mathfrak{S}_P^w = \Delta_k^w/\Delta_k$1 converge precisely to the infinite Toeplitz matrices whose generating functions are exponentials:

$\mathfrak{S}_P^w = \Delta_k^w/\Delta_k$2

with coefficients converging to $\mathfrak{S}_P^w = \Delta_k^w/\Delta_k$3. Conversely, every convergent sequence from $\mathfrak{S}_P^w = \Delta_k^w/\Delta_k$4 has such a limit. An elegant alternative proof uses the functional equation $\mathfrak{S}_P^w = \Delta_k^w/\Delta_k$5, which forces the limit to satisfy $\mathfrak{S}_P^w = \Delta_k^w/\Delta_k$6; combined with the bound $\mathfrak{S}_P^w = \Delta_k^w/\Delta_k$7 (which excludes poles), this implies no zeros or poles, hence — by Edrei's theorem — an exponential.

As a corollary, the authors obtain a Toeplitz-matrix reformulation of the analytic heart of Edrei's theorem: an element of $\mathfrak{S}_P^w = \Delta_k^w/\Delta_k$8 with zero-free, pole-free generating function lies in the limiting set of the middle-dimensional Grassmannian Toeplitz varieties if and only if it is of the form $\mathfrak{S}_P^w = \Delta_k^w/\Delta_k$9. They note explicitly that a direct proof of this corollary would yield a new proof of Edrei's theorem itself; this remains open.

Strange duality within Peterson theory

The paper gives a geometric interpretation of the Postnikov–Hengelbrock strange duality involution $X_P(\mathbb{R}_{>0})$0 on $X_P(\mathbb{R}_{>0})$1, defined by $X_P(\mathbb{R}_{>0})$2. The authors define an involution $X_P(\mathbb{R}_{>0})$3 of the Peterson stratum $X_P(\mathbb{R}_{>0})$4 by $X_P(\mathbb{R}_{>0})$5 — an asymmetric-looking formula pairing the big cell with the opposite Bruhat cell — and prove that $X_P(\mathbb{R}_{>0})$6 recovers $X_P(\mathbb{R}_{>0})$7 under Peterson's isomorphism. The proof establishes the identity $X_P(\mathbb{R}_{>0})$8, with the sign fixed by positivity, and connects to the Chaput–Manivel–Perrin formula $X_P(\mathbb{R}_{>0})$9. A corollary gives the explicit factorisation identity

$Gr(k,n)$00

which also clarifies the meaning of the coordinates $Gr(k,n)$01 on $Gr(k,n)$02: they represent inverted Chern roots of the tautological quotient bundle. This interpretation is then used to give a second, uniform proof of both Schubert class limit theorems via the evaluation formula $Gr(k,n)$03. The authors note that this construction of $Gr(k,n)$04 is specific to the Grassmannian and does not extend verbatim to other partial flags.

Partial flag varieties: partial results and conjectures

The final section formulates conjectures generalising the Grassmannian results. Conjecture 5.3 predicts that for a fixed dimension vector $Gr(k,n)$05, the limits of sequences from $Gr(k,n)$06 are exactly the matrices with generating function $Gr(k,n)$07 where the Schoenberg parameters are constant on blocks between the $Gr(k,n)$08. A more general conjecture covers pairs $Gr(k,n)$09 of finite or infinite sequences, predicting limits with generating functions $Gr(k,n)$10, together with matching Schubert class limit formulas. What is proved unconditionally is the quantum parameter asymptotics when both sequences are infinite:

$Gr(k,n)$11

under a convergence assumption on the relevant corner minors. The proof reduces to the full flag theorem via Kostant's product formula for $Gr(k,n)$12 in terms of the $Gr(k,n)$13, evaluated on truncations $Gr(k,n)$14. Notably, the proposition requires both sequences to be infinite; the finite cases, where boundary terms like $Gr(k,n)$15 should appear, remain conjectural, though verified in the Grassmannian instances already treated.

Limitations and open questions

Several caveats qualify the results. The inverse of the parametrisation map $Gr(k,n)$16 is not algebraic, so recovering matrices from parameters requires solving polynomial systems without closed-form solutions. The Schubert class limit theorems are proved for $Gr(k,n)$17; the degenerate cases require separate treatment. For partial flag varieties, the existence of convergent sequences with prescribed block-constant Schoenberg parameters (Conjecture 5.3 and its generalisations) and the Schubert class limit formulas (Conjecture 5.7) remain open, as do the finite-sequence cases of the quantum parameter asymptotics. Finally, the proposed direct proof of Edrei's theorem via Corollary 4.4 — bypassing Nevanlinna theory entirely — is identified but not carried out.

Conclusion

The paper completes, for Grassmannians, the program initiated in the full flag setting: it determines exactly which infinite totally nonnegative Toeplitz matrices arise as limits of totally positive Peterson Toeplitz points — polynomials $Gr(k,n)$18 for fixed codimension, reciprocals $Gr(k,n)$19 for fixed rank, and exponentials $Gr(k,n)$20 in the balanced case — and computes the corresponding Schubert class limits as Schur polynomial evaluations. The identification of strange duality with a transposition involution on the Peterson variety provides a conceptually satisfying account of a previously ad hoc symmetry, and the partial flag conjectures indicate a uniform Lie-theoretic picture in which the Schoenberg parameters of the limit are dictated blockwise by the parabolic data.